# Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of 3 Qubits: One Reports to Two

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## Abstract

**:**

## 1. Introduction

## 2. QSOD Circuits–3 Qubit Organizational Design Configurations–One Reports to Two

#### 2.1. Quantum Circuit–Case I–Agents B and C Have No Communication between Each Other

$|{\Psi}_{A}\rangle ={a}_{0}|0\rangle +{a}_{1}|1\rangle $ | $\hspace{1em}{a}_{i}\in {\mathbb{C}}^{2}$ |

$|{\Psi}_{B}\rangle ={b}_{0}|0\rangle +{b}_{1}|1\rangle $ | $\hspace{1em}{b}_{i}\in {\mathbb{C}}^{2}$ |

$|{\Psi}_{C}\rangle ={c}_{0}|0\rangle +{c}_{1}|1\rangle $ | $\hspace{1em}{c}_{i}\in {\mathbb{C}}^{2}$. |

- $1-{z}_{1}=P(A=|0\rangle )=1-P(A=|1\rangle )$. Probability of alignment of node A.
- ${x}_{1}=P(B=|1\rangle |A=|0\rangle )$. Probability of no–alignment of node B conditioned to the state of alignment of node A.
- ${y}_{1}=P(B=|1\rangle |A=|1\rangle )$. Probability of no–alignment of node B conditioned to the state of no–alignment of node A.
- ${x}_{2}=P(C=|1\rangle |A=|0\rangle )$. Probability of no–alignment of node C conditioned to the state of alignment of node A.
- ${y}_{2}=P(C=|1\rangle |A=|1\rangle )$. Probability of no–alignment of node C conditioned to the state of no–alignment of node A.

#### 2.2. Quantum Circuit–Case II–Agents B and C Have Communication between Each Other

$|{\Psi}_{A}\rangle ={a}_{0}|0\rangle +{a}_{1}|1\rangle $ | $\hspace{1em}{a}_{i}\in {\mathbb{C}}^{2}$ |

$|{\Psi}_{B}\rangle ={b}_{0}|0\rangle +{b}_{1}|1\rangle $ | $\hspace{1em}{b}_{i}\in {\mathbb{C}}^{2}$ |

$|{\Psi}_{C}\rangle ={c}_{0}|0\rangle +{c}_{1}|1\rangle $ | $\hspace{1em}{c}_{i}\in {\mathbb{C}}^{2}$ |

$|{\Psi}^{*}\rangle ={d}_{0}|0\rangle +{d}_{1}|1\rangle $ | $\hspace{1em}{d}_{i}\in {\mathbb{C}}^{2}$. |

- $1-{z}_{11}=P(A=|0\rangle )=1-P(A=|1\rangle )$. Probability of alignment of node A.
- ${z}_{21}=P(B=|1\rangle |A=|1\rangle )$. Probability of no–alignment of node B conditioned to the state of no–alignment of node A.
- ${z}_{22}=P(B=|1\rangle |A=|0\rangle )$. Probability of no–alignment of node B conditioned to the state of alignment of node A.
- ${x}_{11}=P(C=|1\rangle |A,B=|11\rangle )$. Probability of no–alignment of node C conditioned to the state $|11\rangle $ of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $.
- ${y}_{11}=P(C=|1\rangle |A,B=|10\rangle )$. Probability of no–alignment of node C conditioned to the state $|10\rangle $ of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $.
- ${x}_{21}=P(C=|1\rangle |A,B=|00\rangle )$. Probability of no–alignment of node C conditioned to the state $|00\rangle $ of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $.
- ${y}_{21}=P(C=|1\rangle |A,B=|01\rangle )$. Probability of no–alignment of node C conditioned to the state $|01\rangle $ of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $.

## 3. Case Study

#### 3.1. Simulation–Case I–Agents B and C Have No Communication between Each Other

#### 3.2. Simulation–Case II–Agents B and C Have Communication between Each Other

## 4. Discussion

**R**obtained from the simulations.

#### 4.1. Discussion–Case I–Agents B and C Have No Communication between Each Other

**R1**. Agents B and C have an antagonistic alignment probability. The two never have a high probability of alignment simultaneously. In Figure 2 we can see how, for both high and low values of alignment for node A, $P(A=|0\rangle )=0.9$ or $P(A=|0\rangle )=0.1$ respectively, the alignment probabilities of agents B and C have a negative correlation. When one of the two has high alignment probabilities, the other has low ones.**R2**. Agents B and C only agree by chance. In Figure 2 we can see how, as agent A approaches its random alignment probability of 50%, the alignment probabilities of B and C become homogeneous until reaching the 50% value as well.**R3**. Quantum phase transition with 90% alignment probability of node A. The representations of Figure 3 are particular cases of the general solution of Figure 2. In both we can observe a sharp change of slope of the regression between the alignment probabilities of B and C. This clearly indicates a quantum phase change at the point where the probability of non–alignment of agent A is 10%, $P(A=|1\rangle )=0.1$. In more detail, the observed results show:- -
- As shown in Figure 3, if the alignment probability of A is very high, $P(A=|0\rangle )>0.9$ (or $P(A=|1\rangle )<0.1$), and the probability that B and C are in non–alignment, provided that A is in non–alignment, are equal, ${y}_{1}=P(B=|1\rangle |A=|1\rangle )={y}_{2}=P(C=|1\rangle |A=|1\rangle )$, then the alignment probability of C is very low and does not vary with the alignment probability of B;
- -
- As shown in Figure 3, if the alignment probability of A is not high, $0.15<P(A=|1\rangle )<0.90$, and the probability that B and C are in non–alignment, provided that A is in non–alignment, are equal, ${y}_{1}=P(B=|1\rangle |A=|1\rangle )={y}_{2}=P(C=|1\rangle |A=|1\rangle )$, then the alignment probability of B and C present a positive correlation.

#### 4.2. Discussion–Case II–Agents B and C Have Communication between Each Other

**R5**. Agents B and C interchange energy. Lowering the probability of alignment of node B, $P(B=|0\rangle )$, which can be understood as its energy, while maintaining $P(A=|1\rangle )\in [0.01,0.1]$, shows how $P(C=|0\rangle )$ behaves with changing ${x}_{11}={x}_{21}={x}_{21}={y}_{21}={z}_{21}={z}_{22}$. The curves shown quantify this interaction.

## 5. Conclusions, Limitations and Further Steps

#### 5.1. Conclusions–Case I–Agents B and C Have No Communication between Each Other

**M1**. The management conclusion derived from

**R1**for subordinate agent A is staggering and somehow tragic: If the two bosses do not communicate with each other, A will never be able to serve them in such a way that both are simultaneously in alignment. It does not matter what A does. This could lead one to believe that agent A’s motivation to provide a contribution to the value chain may be diminished due to the very organizational structure in which they are immersed, regardless of capabilities, skills, or attitudes. The organizational design would therefore impose undesirable boundary conditions for the adequate development of the activity of the subordinate node.

**M2**. The conclusions derived from the

**R2**result are not very encouraging for management either. In case the two superior agents do not communicate between them, their joint alignment is always around the point of equilibrium, which is the probability given by the chance. As long as the subordinate node has a higher or lower probability of alignment, their positions will be more or less differentiated. This would imply that the node would tend not to position itself with either of the two nodes to which it reports and the expected behavior on its part would be one of a lack of decision-making that could potentially jeopardize the efficiency of the associated value creation processes.

**M3**. The conclusions derived from the

**R3**confirm the results obtained in [20]: Only a strong alignment probability at lower reporting levels enables alignment at higher levels. It seems that empirically the threshold is set by 90%. To grow the organizational network towards strategic objectives, it is necessary to ensure asymptotic stability at the operational levels of the organization. These lower levels are generally the levels closest to the creation of value and it seems logical that they are the sustaining base of the organizational structure.

#### 5.2. Conclusions–Case II–Agents B and C Have Communication between Each Other

**M4**. The conclusions derived from the

**R4**result is that high levels of alignment in both reporting agents A and B do not imply a high level of alignment of node C. In the case in which B and C high levels of alignment in node C are only attained for an entangled system in which A, B, and C are highly dependent, given the condition ${x}_{11}={x}_{21}={x}_{21}={y}_{21}={z}_{21}={z}_{22}all\in ]0,0.1]\cup [0.9,1[$.

**M5**. The conclusions derived from

**R5**show that the interaction between the superior agents B and C becomes manifest when the alignment probability of A is fixed at values higher than 90%. Both superior agents B and C present a non-linear interaction, and depending on what agent should be prioritized, strategies can be then taken towards one or toward other.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

QSOD | Quantum Strategic Organizational Design |

KPI | Key Performance Indicators |

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**Figure 1.**Quantum simulation of Strategic Organizational Design (QSOD). Case of three qubits configuration in which one node reports to two. I. Without communication between the leaders. II. With communication between the leaders.

**Figure 2.**Correlation between $P(B=|0\rangle )$ and $P(C=|0\rangle )$ for different values of ${z}_{1}=P(A=|1\rangle )\in {\xi}_{1}$ for the case of no communication between B and C.

**Figure 3.**Correlation between $P(B=|0\rangle )$ and $P(C=|0\rangle )$ for different values of ${y}_{1}=P(B=|1\rangle |A=|1\rangle )={y}_{2}=P(C=|1\rangle |A=|1\rangle )\in {\xi}_{2}$ for the case of no communication between B and C.

**Figure 5.**Alignment Probabilities of $P(A=|0\rangle ),P(B=|0\rangle )$ and $P(C=|0\rangle )$ with ${z}_{11}=P(A=|1\rangle )\in [0.01,0.1]$ for different values of fixed ${z}_{21}=P(B=|1\rangle |A=|1\rangle )={z}_{22}=P(B=|1\rangle |A=|0\rangle )$, and combinations of ${x}_{11}={x}_{21}={x}_{21}={y}_{21}$.

Qubit | Interpretation | Equation |
---|---|---|

$|{\Psi}_{A}\rangle $ | The probability ${z}_{1}=P(A=|1\rangle )$ of qubit $|{\Psi}_{A}\rangle $ to be in not–alignment translates into the rotation angle ${\theta}_{{z}_{1}}$. | ${\theta}_{{z}_{1}}=2arctan\sqrt{\frac{{z}_{1}}{1-{z}_{1}}}$ |

$|{\Psi}_{B}\rangle $ | The conditional probability ${x}_{1}=P(B=|1\rangle |A=|0\rangle )$ of qubit $|{\Psi}_{B}\rangle $ to be in not–alignment depending on the probability of $|{\Psi}_{A}\rangle $ to be in the state $|0\rangle $ translates into rotation angle ${\theta}_{{x}_{1}}$. | ${\theta}_{{x}_{1}}=2arctan\sqrt{\frac{{x}_{1}}{1-{x}_{1}}}$ |

The conditional probability ${y}_{1}=P(B=|1\rangle |A=|1\rangle )$ of qubit $|{\Psi}_{B}\rangle $ to be in not–alignment depending on the probability of $|{\Psi}_{A}\rangle $ to be in the state $|1\rangle $ translates into rotation angle ${\theta}_{{y}_{1}}$. | ${\theta}_{{y}_{1}}=2arctan\sqrt{\frac{{y}_{1}}{1-{y}_{1}}}$ | |

$|{\Psi}_{C}\rangle $ | The conditional probability ${x}_{2}=P(C=|1\rangle |A=|0\rangle )$ of qubit $|{\Psi}_{C}\rangle $ to be in not–alignment depending on the probability of $|{\Psi}_{A}\rangle $ to be in the state $|0\rangle $ translates into rotation angle ${\theta}_{{x}_{2}}$. | ${\theta}_{{x}_{2}}=2arctan\sqrt{\frac{{x}_{2}}{1-{x}_{2}}}$ |

The conditional probability ${y}_{2}=P(C=|1\rangle |A=|1\rangle )$ of qubit $|{\Psi}_{C}\rangle $ to be in not–alignment depending on the probability of $|{\Psi}_{A}\rangle $ to be in the state $|1\rangle $ translates into rotation angle ${\theta}_{{y}_{2}}$. | ${\theta}_{{y}_{2}}=2arctan\sqrt{\frac{{y}_{2}}{1-{y}_{2}}}$ |

Qubit | Interpretation | Equation |
---|---|---|

$|{\Psi}_{A}\rangle $ | The probability ${z}_{11}=P(A=|1\rangle )$ of qubit $|{\Psi}_{A}\rangle $ to be in not–alignment translates into the rotation angle ${\theta}_{{z}_{11}}$. | ${\theta}_{{z}_{11}}=2arctan\sqrt{\frac{{z}_{11}}{1-{z}_{11}}}$ |

$|{\Psi}_{B}\rangle $ | The conditional probability ${z}_{21}=P(B=|1\rangle |A=|1\rangle )$ of qubit $|{\Psi}_{B}\rangle $ to be in not–alignment depending on the probability of $|{\Psi}_{A}\rangle $ to be in the state $|1\rangle $ translates into rotation angle ${\theta}_{{z}_{21}}$. | ${\theta}_{{z}_{21}}=2arctan\sqrt{\frac{{z}_{21}}{1-{z}_{21}}}$ |

The conditional probability ${z}_{22}=P(B=|1\rangle |A=|0\rangle )$ of qubit $|{\Psi}_{B}\rangle $ to be in not–alignment depending on the probability of $|{\Psi}_{A}\rangle $ to be in the state $|0\rangle $ translates into rotation angle ${\theta}_{{z}_{22}}$. | ${\theta}_{{z}_{22}}=2arctan\sqrt{\frac{{z}_{22}}{1-{z}_{22}}}$ | |

$|{\Psi}_{C}\rangle $ | The conditional probability ${x}_{11}=P(C=|1\rangle |A,B=|11\rangle )$ of qubit $|{\Psi}_{C}\rangle $ to be in not–alignment depending on the probability of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $ to be in the state $|11\rangle $ translates into rotation angle ${\theta}_{{x}_{1}1}$. | ${\theta}_{{x}_{11}}=2arctan\sqrt{\frac{{x}_{11}}{1-{x}_{11}}}$ |

The conditional probability ${y}_{11}=P(C=|1\rangle |A,B=|10\rangle )$ of qubit $|{\Psi}_{C}\rangle $ to be in not–alignment depending on the probability of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $ to be in the state $|10\rangle $ translates into rotation angle ${\theta}_{{y}_{11}}$. | ${\theta}_{{y}_{11}}=2arctan\sqrt{\frac{{y}_{11}}{1-{y}_{11}}}$ | |

The conditional probability ${x}_{21}=P(C=|1\rangle |A,B=|00\rangle )$ of qubit $|{\Psi}_{C}\rangle $ to be in not–alignment depending on the probability of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $ to be in the state $|00\rangle $ translates into rotation angle ${\theta}_{{x}_{2}1}$. | ${\theta}_{{x}_{21}}=2arctan\sqrt{\frac{{x}_{21}}{1-{x}_{21}}}$ | |

The conditional probability ${y}_{21}=P(C=|1\rangle |A,B=|01\rangle )$ of qubit $|{\Psi}_{C}\rangle $ to be in not–alignment depending on the probability of the waveform $|{\Psi}_{A}\rangle \otimes |{\Psi}_{B}\rangle $ to be in the state $|01\rangle $ translates into rotation angle ${\theta}_{{y}_{21}}$. | ${\theta}_{{y}_{21}}=2arctan\sqrt{\frac{{y}_{21}}{1-{y}_{21}}}$ | |

$|{\Psi}^{*}\rangle $ | The ancilla qubit $|{\Psi}^{*}\rangle $ is a support qubit and as such is not subject to any conditional probability rotation. |

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Villalba-Diez, J.; Losada, J.C.; Benito, R.M.; González-Marcos, A.
Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of 3 Qubits: One Reports to Two. *Entropy* **2021**, *23*, 374.
https://doi.org/10.3390/e23030374

**AMA Style**

Villalba-Diez J, Losada JC, Benito RM, González-Marcos A.
Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of 3 Qubits: One Reports to Two. *Entropy*. 2021; 23(3):374.
https://doi.org/10.3390/e23030374

**Chicago/Turabian Style**

Villalba-Diez, Javier, Juan Carlos Losada, Rosa María Benito, and Ana González-Marcos.
2021. "Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of 3 Qubits: One Reports to Two" *Entropy* 23, no. 3: 374.
https://doi.org/10.3390/e23030374